Imperial College London


Faculty of Natural SciencesDepartment of Mathematics

EPSRC Postdoctoral Research Fellow







614Huxley BuildingSouth Kensington Campus





Publication Type

11 results found

Feyzbakhsh S, 2022, An effective restriction theorem via wall-crossing and Mercat's conjecture, MATHEMATISCHE ZEITSCHRIFT, Vol: 301, Pages: 4175-4199, ISSN: 0025-5874

Journal article

Feyzbakhsh S, Thomas RP, 2022, Rank r DT theory from rank 1, Journal of the American Mathematical Society, ISSN: 0894-0347

Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture ofBayer-Macr`ı-Toda, such as the quintic 3-fold.We express Joyce’s generalised DT invariants counting Gieseker semistable sheaves ofany rank r on X in terms of those counting sheaves of rank 1. By the MNOP conjecturethey are therefore determined by the Gromov-Witten invariants of X.

Journal article

Feyzbakhsh S, Li C, 2021, Higher rank Clifford indices of curves on a K3 surface, SELECTA MATHEMATICA-NEW SERIES, Vol: 27, ISSN: 1022-1824

Journal article

Feyzbakhsh S, 2020, Mukai’s program (reconstructing a K3 surface from a curve) via wall-crossing, Journal für die reine und angewandte Mathematik, Vol: 2020, Pages: 101-137, ISSN: 0075-4102

Let C be a curve of genus g=11 or g≥13 on a K3 surface whose Picard group is generated by the curve class [C]. We use wall-crossing with respect to Bridgeland stability conditions to generalise Mukai’s program to this situation: we show how to reconstruct the K3 surface containing the curve C as a Fourier–Mukai transform of a Brill–Noether locus of vector bundles on C.

Journal article

Feyzbakhsh S, Thomas RP, 2019, An application of wall-crossing to Noether-Lefschetz loci, Publisher: arXiv

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov-Giesekerconjecture of Bayer-Macr\`{i}-Toda (such as $\mathbb P^3$, the quinticthreefold or an abelian threefold). Let $L$ be a line bundle supported on a very positive surface in $X$. If$c_1(L)$ is a primitive cohomology class then we show it has very negativesquare.

Working paper

Feyzbakhsh S, Explicit formulae for rank zero DT invariants and the OSV conjecture

Fix a Calabi-Yau 3-fold $X$ satisfying the Bogomolov-Gieseker conjecture ofBayer-Macr\`i-Toda, such as the quintic 3-fold. By two different wall-crossingarguments we prove two different explicit formulae relating rank 0Donaldson-Thomas invariants (counting torsion sheaves on $X$ supported on ampledivisors) in terms of rank 1 Donaldson-Thomas invariants (counting idealsheaves of curves) and Pandharipande-Thomas invariants. In particular, we provea slight modification of Toda's formulation of OSV conjecture for $X$. When $X$is of Picard rank one, we also give an explicit formula for rank two DTinvariants in terms of rank zero and rank one DT invariants.

Journal article

Feyzbakhsh S, Pertusi L, Serre-invariant stability conditions and Ulrich bundles on cubic threefolds, \'Epijournal de G\'eom\'etrie Alg\'ebrique, Volume 7 (2023), Article No. 1

We prove a general criterion which ensures that a fractional Calabi--Yaucategory of dimension $\leq 2$ admits a unique Serre-invariant stabilitycondition, up to the action of the universal cover of$\text{GL}^+_2(\mathbb{R})$. We apply this result to the Kuznetsov component$\text{Ku}(X)$ of a cubic threefold $X$. In particular, we show that all theknown stability conditions on $\text{Ku}(X)$ are invariant with respect to theaction of the Serre functor and thus lie in the same orbit with respect to theaction of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As anapplication, we show that the moduli space of Ulrich bundles of rank $\geq 2$on $X$ is irreducible, answering a question asked by Lahoz, Macr\`i andStellari.

Journal article

Bayer A, Beentjes S, Feyzbakhsh S, Hein G, Martinelli D, Rezaee F, Schmidt Bet al., The desingularization of the theta divisor of a cubic threefold as a moduli space

We show that the moduli space $\overline{M}_X(v)$ of Gieseker stable sheaveson a smooth cubic threefold $X$ with Chern character $v = (3,-H,-H^2/2,H^3/6)$is smooth and of dimension four. Moreover, the Abel-Jacobi map to theintermediate Jacobian of $X$ maps it birationally onto the theta divisor$\Theta$, contracting only a copy of $X \subset \overline{M}_X(v)$ to thesingular point $0 \in \Theta$. We use this result to give a new proof of a categorical version of theTorelli theorem for cubic threefolds, which says that $X$ can be recovered fromits Kuznetsov component $\operatorname{Ku}(X) \subset\mathrm{D}^{\mathrm{b}}(X)$. Similarly, this leads to a new proof of thedescription of the singularity of the theta divisor, and thus of the classicalTorelli theorem for cubic threefolds, i.e., that $X$ can be recovered from itsintermediate Jacobian.

Journal article

Feyzbakhsh S, Hyperkähler varieties as Brill-Noether loci on curves

Consider the moduli space $M_C(r; K_C)$ of stable rank r vector bundles on acurve $C$ with canonical determinant, and let $h$ be the maximum number oflinearly independent global sections of these bundles. If $C$ embeds in a K3surface $X$ as a generator of $Pic(X)$ and the genus $g$ of $C$ is sufficientlyhigh, we show the Brill-Noether locus $BN_C \subset M_C(r; K_C)$ of bundleswith $h$ global sections is a smooth projective Hyperk\"{a}hler manifold ofdimension $2g -2r \lfloor \frac{g}{r}\rfloor$, isomorphic to a moduli space ofstable vector bundles on $X$.

Journal article

Feyzbakhsh S, Mukai's Program (reconstructing a K3 surface from a curve) via wall-crossing, II

Let $C$ be a curve on a K3 surface $X$ with Picard group $\mathbb{Z}.[C]$.Mukai's program seeks to recover $X$ from $C$ by exhibiting it as aFourier-Mukai partner to a Brill-Noether locus of vector bundles on $C$. We usewall-crossing in the space of Bridgeland stability conditions to prove this forgenus $\ge14$. This paper deals with the case $g-1$ prime left over from PaperI.

Journal article

Feyzbakhsh S, Thomas RP, Curve counting and S-duality

We work on a projective threefold $X$ which satisfies the Bogomolov-Giesekerconjecture of Bayer-Macr\`i-Toda, such as $\mathbb P^3$ or the quinticthreefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on $X$ aresmooth bundles over Hilbert schemes of ideal sheaves of curves and points in$X$. When $X$ is Calabi-Yau this gives a simple wall crossing formula expressingcurve counts (and so ultimately Gromov-Witten invariants) in terms of counts ofD4-D2-D0 branes. These latter invariants are predicted to have modularproperties which we discuss from the point of view of S-duality andNoether-Lefschetz theory.

Journal article

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